Asymptotic likelihood of chaos for smooth families of circle maps

TL;DR

This paper analyzes the likelihood of chaos in smooth families of circle maps with large parameters, showing that the set of parameters leading to exponential derivative growth approaches full measure as the parameter increases.

Contribution

It improves previous results by providing an asymptotic estimate that the measure of parameters inducing chaos tends to full measure as the parameter L grows large.

Findings

The measure of parameters with exponential growth of derivatives approaches 1 as L increases.

The construction significantly refines earlier work by Wang and Young.

The results apply to a broad class of circle maps with critical points.

Abstract

We consider a smooth two-parameter family $f_{a,L}\colon\theta\mapsto \theta+a+L\Phi(\theta)$ of circle maps with a finite number of critical points. For sufficiently large $L$ we construct a set $A_L^{(\infty)}$ of $a$-values of positive Lebesgue measure for which the corresponding $f_{a,L}$ exhibits an exponential growth of derivatives along the orbits of the critical points. Our construction considerably improves the previous one of Wang and Young for the same class of families, in that the following asymptotic estimate holds: the Lebesgue measure of $A_L^{(\infty)}$ tends to full measure in $a$-space as $L$ tends to infinity.